Rounding and EstimationAQA GCSE Maths: Revision notes
Section 1
How do you round to decimal places and significant figures?
Rounding to decimal places (d.p.) means keeping a set number of digits after the decimal point.
- Identify the digit in the position you're rounding to
- Look at the digit immediately to the right
- If it is 5 or more, round up; if less than 5, round down
- Remove all digits to the right of your rounding position
Rounding to significant figures (s.f.) counts all digits that carry meaning, starting from the first non-zero digit.
- The first significant figure is the first non-zero digit
- All subsequent digits (including zeros between non-zero digits) count as significant figures
- Trailing zeros after the decimal point are significant; trailing zeros before the decimal point may not be (use context)
| Number | 1 s.f. | 2 s.f. | 3 s.f. |
|---|---|---|---|
| 24,567 | 20,000 | 25,000 | 24,600 |
| 0.004829 | 0.005 | 0.0048 | 0.00483 |
| 156.8 | 200 | 160 | 157 |
Both methods round in the same way—the difference is what counts as a significant figure.
When rounding to significant figures, the position of the decimal point changes—you might end up writing 0.005 or 20,000. The examiner wants to see you count significant figures correctly from the first non-zero digit, regardless of where the decimal point sits.
Round 0.06284 to 2 s.f. The first significant figure is 6, the second is 2. Look at 8 (the next digit)—it's ≥5, so round up: 0.063. Note: the leading zeros do not count as significant figures.
Section 2
How do you estimate answers using one significant figure?
Estimation involves rounding all numbers in a calculation to one significant figure, then performing the calculation with these rounded values. This gives a quick approximation of the answer without a calculator.
Steps for estimation:
- Round each number to 1 s.f.
- Perform the calculation using the rounded values
- Write your estimated answer
Why estimate?
- Check whether a calculated answer is sensible
- Quick mental arithmetic in exams
- Identify calculation errors (if your calculator answer is very different from your estimate, something is wrong)
Example calculations:
- 23.8 × 4.2 ≈ 20 × 4 = 80 (actual: 99.96)
- 187 ÷ 3.2 ≈ 200 ÷ 3 ≈ 67 (actual: 58.4)
- 4.85 + 12.3 + 8.91 ≈ 5 + 10 + 9 = 24 (actual: 26.06)
Always show both the rounded numbers and the calculation. The examiner wants to see your working so they can award method marks even if your mental arithmetic is slightly off.
Show your rounding step clearly: write something like '23.8 ≈ 20' before doing the calculation. Examiners give method marks for correct rounding even if your mental arithmetic has a small slip.
Students often forget to round before calculating, or round only some numbers. You must round every number to 1 s.f. before you do any arithmetic.
Think of rounding to 1 s.f. as zooming out on a map—you lose detail but get a quick sense of direction and distance. It's a 'rough and ready' check, not meant to be precise.
Section 3
What are error intervals and how do you write them using inequality notation?
When a number is rounded or truncated, the true value could be anywhere within a range. This range is called the error interval, and it must include the rounded value at one end but exclude it at the other.
For rounding:
If a number is rounded to a certain degree of accuracy, the true value lies within an interval:
- Rounded value = 12 (to the nearest whole number)
- Lower bound = 11.5, upper bound = 12.5
- Error interval: 11.5 ≤ x < 12.5
Note: the lower bound is included (≤), but the upper bound is excluded (<).
For truncation:
Truncation means cutting off digits without rounding. The true value is always greater than the truncated value but less than the next possible value.
- Truncated value = 3.4 (to 1 d.p.)
- Error interval: 3.4 ≤ x < 3.5
General rule for error intervals:
| Rounded/truncated to | Lower bound | Upper bound | Inequality |
|---|---|---|---|
| Nearest integer | value − 0.5 | value + 0.5 | lower ≤ x < upper |
| 1 d.p. | value − 0.05 | value + 0.05 | lower ≤ x < upper |
| 1 s.f. | depends on the digit size | — | — |
| 10 | value − 5 | value + 5 | lower ≤ x < upper |
Key point: the lower bound is always included (≤), and the upper bound is always excluded (<). This ensures each true value falls into exactly one interval.
Always use ≤ for the lower bound and < for the upper bound. This is standard notation—examiners expect it. If you use ≥ or ≤ for both, you will lose marks.
A mass is recorded as 5.2 kg to 1 d.p. Find the error interval. The true mass could be from 5.15 kg up to (but not including) 5.25 kg. Write: 5.15 ≤ m < 5.25
Students often write both bounds using ≤ or both using <. Remember: lower bound uses ≤ (included), upper bound uses < (excluded). This ensures no overlap between consecutive intervals.
Section 4
What are upper and lower bounds, and how do you use them in calculations?
Upper and lower bounds are the maximum and minimum possible values of a measurement given its degree of accuracy. They are identical to the error interval endpoints, but are used actively in calculations.
Finding bounds for a single measurement:
If a length is measured as 8.5 cm to 1 d.p.:
- Lower bound = 8.45 cm (the smallest value that rounds to 8.5)
- Upper bound = 8.55 cm (the smallest value that would round to 8.6, but doesn't reach it)
Using bounds in calculations:
When measurements with bounds are combined (added, multiplied, etc.), the result also has bounds.
For addition or multiplication (to find maximum result): use upper bounds for all measurements
For subtraction (larger − smaller) or division (larger ÷ smaller) to find maximum result: use the upper bound of the numerator and lower bound of the denominator
For subtraction (larger − smaller) or division to find minimum result: use the lower bound of the numerator and upper bound of the denominator
Practical example:
A rectangle has length 5.3 cm and width 3.8 cm, both to 1 d.p. Find the range of possible areas.
- Lower bounds: L = 5.25, W = 3.75
- Upper bounds: L = 5.35, W = 3.85
- Minimum area = 5.25 × 3.75 = 19.6875 cm²
- Maximum area = 5.35 × 3.85 = 20.5975 cm²
- Range: 19.6875 cm² ≤ A < 20.5975 cm² (or round sensibly for presentation)
Always state which bounds you used and why. The examiner wants to see your reasoning.
For multiplication or addition, always multiply/add the upper bounds to get the maximum, and the lower bounds to get the minimum. For division, flip: use upper bound on top and lower bound on bottom for the maximum result.
A distance is 12.5 m (to 1 d.p.) and time is 4.2 s (to 1 d.p.). Speed = distance ÷ time. Bounds: L(d)=12.45, U(d)=12.55, L(t)=4.15, U(t)=4.25. Max speed = 12.55 ÷ 4.15 ≈ 3.02 m/s. Min speed = 12.45 ÷ 4.25 ≈ 2.93 m/s.
For division, students often forget to flip and use upper/lower bounds incorrectly. Remember: to maximise a fraction, maximise the numerator and minimise the denominator. To minimise, do the opposite.
Section 5
How do limits of accuracy apply to real-world measurements?
Limits of accuracy describe the range of possible true values for any measurement or rounded number. They are the same as bounds and error intervals, but the term emphasises the practical context.
Why limits of accuracy matter:
- All real measurements have some uncertainty due to measuring instruments and rounding
- When measurements are used in formulas or further calculations, this uncertainty propagates
- Engineers, scientists, and manufacturers must account for these limits to ensure products are safe and fit for purpose
Applying limits of accuracy:
- Identify the degree of accuracy of each measurement (nearest 1 cm, to 1 d.p., to 1 s.f., etc.)
- Calculate the bounds of each measurement
- Use bounds in the formula to find the range of possible results
- Interpret the result in context: is the answer acceptable for the application?
Real-world context:
If a concrete beam is designed to support a load and its dimensions are given as 2.5 m (to 1 d.p.), the actual beam could be between 2.45 m and 2.55 m long. Engineers must calculate that even the smallest or largest possible beam will perform safely. The same logic applies to medicine dosages, manufacturing tolerances, and safety factors.
Presentation in exams:
Always clearly state:
- The bounds or error interval for each measurement
- How you combined the bounds (which bounds you used for max/min)
- The final range or limits of the calculated quantity
- A brief comment on what the result means in context if asked
When a question asks about limits of accuracy in context (e.g. 'Will the component fit?'), always check both the minimum and maximum possible values. Show that you understand real measurements have unavoidable uncertainty.
A square tile has side length 20 cm (to the nearest cm). Find the range of possible areas. Bounds: 19.5 ≤ s < 20.5. Min area = 19.5² = 380.25 cm². Max area = 20.5² = 420.25 cm². Range: 380.25 ≤ A < 420.25 cm².
Must Know
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Rounding to decimal places: count digits after the decimal point; to significant figures, count from the first non-zero digit. Use 5 or more to round up, less than 5 to round down.
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Estimation: round all numbers to 1 s.f., then calculate. Show your rounding step clearly to gain method marks.
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Error intervals use inequality notation: lower bound ≤ x < upper bound. The lower bound is included (≤), the upper bound is excluded (<). This applies to both rounding and truncation.
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Bounds in calculations: for maximum results, use all upper bounds (or upper bound of numerator, lower bound of denominator in division). For minimum results, reverse this. Always state which bounds you used.
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Upper and lower bounds are the extreme possible values of a measurement. For a rounded value, the lower bound is the value itself and the upper bound is the next rounding threshold. The degree of accuracy determines the bound interval (±0.5 for nearest integer, ±0.05 for 1 d.p., etc.).
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Limits of accuracy apply to real-world measurements and must be accounted for in all calculations involving measured quantities. Always present bounds clearly and interpret results in context when asked.
That's the notes covered.
Carry on to the next subtopic.